| Min(phi) over symmetries of the knot is: [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,1,1,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.1579', '7.40958'] |
| Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.239', '6.428', '6.470', '6.556', '6.700', '6.910', '6.962', '6.1006', '6.1013', '6.1038', '6.1207', '6.1224', '6.1225', '6.1269', '6.1270', '6.1308', '6.1319', '6.1320', '6.1323', '6.1485', '6.1551', '6.1579', '6.1581', '6.1660', '6.1672', '6.1679', '6.1711', '6.1719', '6.1732', '6.1745', '6.1748', '6.1827', '6.1836', '6.1838', '6.1850', '6.1866'] |
| Outer characteristic polynomial of the knot is: t^7+20t^5+22t^3+4t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1579', '7.40958'] |
| 2-strand cable arrow polynomial of the knot is: -64*K1**6 + 448*K1**4*K2 - 1312*K1**4 + 96*K1**3*K2*K3 - 768*K1**3*K3 - 816*K1**2*K2**2 - 160*K1**2*K2*K4 + 3568*K1**2*K2 - 288*K1**2*K3**2 - 64*K1**2*K4**2 - 2604*K1**2 - 128*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2272*K1*K2*K3 + 712*K1*K3*K4 + 96*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 48*K2**2*K4**2 + 216*K2**2*K4 - 1788*K2**2 + 64*K2*K3*K5 + 32*K2*K4*K6 - 904*K3**2 - 318*K4**2 - 36*K5**2 - 4*K6**2 + 1932 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1579'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4218', 'vk6.4299', 'vk6.5479', 'vk6.5592', 'vk6.7585', 'vk6.7681', 'vk6.9087', 'vk6.9168', 'vk6.11172', 'vk6.12254', 'vk6.12363', 'vk6.19371', 'vk6.19664', 'vk6.19776', 'vk6.26153', 'vk6.26211', 'vk6.26569', 'vk6.26654', 'vk6.30770', 'vk6.31291', 'vk6.31688', 'vk6.31971', 'vk6.32445', 'vk6.32862', 'vk6.38157', 'vk6.38199', 'vk6.39095', 'vk6.41349', 'vk6.44818', 'vk6.44940', 'vk6.45847', 'vk6.48528', 'vk6.49334', 'vk6.52307', 'vk6.53147', 'vk6.58449', 'vk6.62969', 'vk6.63586', 'vk6.66313', 'vk6.66355'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3U4O5U2U5O6O4U1U3U6 |
| R3 orbit | {'O1O2U3O4O5U2U5O3O6U1U6U4', 'O1O2O3U4O5U2U5O6O4U1U3U6'} |
| R3 orbit length | 2 |
| Gauss code of -K | O1O2O3U4U1U3O5O4U6U2O6U5 |
| Gauss code of K* | O1O2O3U1U4U2O5U6O4O6U3U5 |
| Gauss code of -K* | O1O2O3U4U1O5O6U5O4U2U6U3 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -1 1 0 1 1],[ 2 0 0 2 1 1 1],[ 1 0 0 1 0 1 0],[-1 -2 -1 0 -1 0 0],[ 0 -1 0 1 0 1 1],[-1 -1 -1 0 -1 0 0],[-1 -1 0 0 -1 0 0]] |
| Primitive based matrix | [[ 0 1 1 1 0 -1 -2],[-1 0 0 0 -1 0 -1],[-1 0 0 0 -1 -1 -1],[-1 0 0 0 -1 -1 -2],[ 0 1 1 1 0 0 -1],[ 1 0 1 1 0 0 0],[ 2 1 1 2 1 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-1,-1,-1,0,1,2,0,0,1,0,1,0,1,1,1,1,1,2,0,1,0] |
| Phi over symmetry | [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,1,1,1,0,0,0] |
| Phi of -K | [-2,-1,0,1,1,1,1,1,1,2,2,1,1,1,2,0,0,0,0,0,0] |
| Phi of K* | [-1,-1,-1,0,1,2,0,0,0,1,1,0,0,1,2,0,2,2,1,1,1] |
| Phi of -K* | [-2,-1,0,1,1,1,0,1,1,1,2,0,0,1,1,1,1,1,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | z^2+14z+25 |
| Enhanced Jones-Krushkal polynomial | w^3z^2+14w^2z+25w |
| Inner characteristic polynomial | t^6+12t^4+9t^2+1 |
| Outer characteristic polynomial | t^7+20t^5+22t^3+4t |
| Flat arrow polynomial | -6*K1**2 - 4*K1*K2 + 2*K1 + 3*K2 + 2*K3 + 4 |
| 2-strand cable arrow polynomial | -64*K1**6 + 448*K1**4*K2 - 1312*K1**4 + 96*K1**3*K2*K3 - 768*K1**3*K3 - 816*K1**2*K2**2 - 160*K1**2*K2*K4 + 3568*K1**2*K2 - 288*K1**2*K3**2 - 64*K1**2*K4**2 - 2604*K1**2 - 128*K1*K2**2*K3 - 64*K1*K2*K3*K4 + 2272*K1*K2*K3 + 712*K1*K3*K4 + 96*K1*K4*K5 - 24*K2**4 - 48*K2**2*K3**2 - 48*K2**2*K4**2 + 216*K2**2*K4 - 1788*K2**2 + 64*K2*K3*K5 + 32*K2*K4*K6 - 904*K3**2 - 318*K4**2 - 36*K5**2 - 4*K6**2 + 1932 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {4, 5}, {1, 3}], [{6}, {1, 5}, {3, 4}, {2}]] |
| If K is slice | False |